Given that most of the above references focus on Coxeter systems of particular types, it is perhaps natural to
wonder whether one can systematically study Kazhdan–Lusztig cells of an arbitrary Coxeter system \((W,S)\) . The
\(\mathbf {a}\) -function, a certain function \(\mathbf {a}: W\rightarrow \mathbb {Z}_{\ge 0}\) , offers an important tool in this study. The
\(\mathbf {a}\) -function is due to Lusztig, who first defined it for Weyl and affine Weyl groups in [1]}, [2]} and then extended the definition to general Coxeter groups in [3]}. Lusztig showed
that the \(\mathbf {a}\) -function takes a constant value on every two-sided cell and hence on every left or right cell, so it
makes sense to speak of the \(\mathbf {a}\) -value of a cell, and the set \(W_n:=\lbrace w\in W:\mathbf {a}(w)=n\rbrace \)  is a union of two-sided
cells for all \(n\in \mathbb {Z}_{\ge 0}\) . In principle, one may also hope to organize and study cells by \(\mathbf {a}\) -values.
Indeed, cells of \(\mathbf {a}\) -values 0 and 1 in a general Coxeter system \((W,S)\)  are well understood. For example, if we
assume \((W,S)\)  is irreducible (see rmk:irreducible), then both \(W_0\)  and \(W_1\)  contain a single
two-sided cell, we have \(W_0=\lbrace 1\rbrace \) , and \(W_1\)  consists of all non-identity elements in \(W\)  with a unique reduced
word. Moreover, Coxeter systems where \(W_1\)  is finite have been classified, and the cardinality of \(W_1\)  is known
for such systems. For more details on the cell \(W_1\) , see prop:facts01, [4]},
[5]}, [6]} and [7]}.
